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I've asked a very similar question also at math.stackexchange, but I've not received any answer.

A vectorial function $\boldsymbol{x}:\mathbb{R}^D \rightarrow \mathbb{R}^N$ satisfies the following linear PDE

$$ \sum_{i=1}^D c_i \frac{\partial \boldsymbol{x}(t_1, \dots, t_D)}{\partial t_i} = \boldsymbol{A}(t_1, \dots, t_D) \boldsymbol{x}(t_1, \dots, t_D). $$

The scalars $c_i$ are constant, and $\boldsymbol{A}(t_1, \dots, t_D): \mathbb{R}^D \rightarrow \mathbb{R}^{N \times N}$ is quasiperiodic, i.e.

$$ \boldsymbol{A}(t_1, \dots, t_i+T_i, \dots, t_D) = \boldsymbol{A}(t_1, \dots, t_i, \dots, t_D). $$

Which boundary conditions should I impose in order to numerically find one of its fundamental solution matrices?

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  • $\begingroup$ Since the $c_i$ are constant (as you say over at math.SE), you can choose coordinates $(s_i)$ such that $\partial/\partial s_1 = \sum_i^D c_i \partial/\partial t_i$. Then your PDE is just an ODE in $s_1$ with $D-1$ parameters. This reduces the PDE boundary conditions to $(s_2,\ldots, s_D)$-parametrized ODE initial/boundary conditions. Perhaps this is sufficient for your purposes. $\endgroup$ Jun 6, 2016 at 14:56
  • $\begingroup$ Thank you very much for your help! It's the same that has been written in JA Murdock - On the Floquet Problem for Quasiperiodic Systems. I'll try to think about it. $\endgroup$
    – Jommy
    Jun 7, 2016 at 15:39

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